Consultation Document - edb.gov.hk · PDF fileLearning Objectives of Module 1 ... summation...

Consultation Document

p. 1

Learning Objectives of Module 1 (Calculus and Statistics) Notes:

1. Learning units are grouped under three areas (“Foundation Knowledge”, “Calculus” and “Statistics”) and a Further Learning Unit.

2. Related learning objectives are grouped under the same learning unit.

3. The notes in the “Remarks” column of the table may be considered as supplementary information about the learning objectives.

4. To aid teachers in judging how far to take a given topic, a suggested lesson time in hours is given against each learning unit. However, the lesson time assigned is for their reference only. Teachers may adjust the lesson time to meet their individual needs.

Learning Unit Learning Objective Time Remarks

Foundation Knowledge

1. Binomial expansion

1.1 recognise the expansion of ( )na b , where n is a positive integer

3 Students are required to recognise the summation notation ().

The following content are not required:

expansion of trinomials the greatest coefficient, the greatest

term and the properties of binomial coefficients

applications to numerical approximation


p. 2


2. Exponential and logarithmic functions

2.1 recognise the definition of the number e and the exponential

series 2 3

12! 3!

x x xe x

8

2.2 understand exponential functions and logarithmic functions The following functions are required: xy e lny x

2.3 use exponential functions and logarithmic functions to solve problems

2.4 transform xy ka and ( ) ny k f x to linear relations,

where a , n and k are real numbers, 0a , 1a , ( ) 0f x and ( ) 1f x

Students are required to solve problems including those related to compound interest, population growth and radioactive decay.

When experimental values of x and y are given, students are required to plot the graph of the corresponding linear relation from which they can determine the values of the unknown constants by considering its slope and intercepts.

Subtotal in hours 11


p. 3


Calculus

3. Derivative of a function

3.1 recognise the intuitive concept of the limit of a function

3.2 find the limits of algebraic functions, exponential functions and logarithmic functions

5 Student are required to recognise the theorems on the limits of sum, difference, product, quotient, scalar multiplication of functions and the limits of composite functions (the proofs are not required).

The following algebraic functions are required:

polynomial functions

rational functions

power functions x

functions derived from the above ones through addition, subtraction, multiplication, division and composition, such as 2 1x


p. 4


3.3 recognise the concept of the derivative of a function from first principles

Students are not required to find the derivatives of functions from first principles.

Students are required to recognise the

notations: y , ( )f x and dydx

.

3.4 recognise the slope of the tangent of the curve ( )y f x at a point 0x x


notations: 0'( )f x and 0x x

dydx

.

4. Differentiation of a function

4.1 understand the addition rule, product rule, quotient rule and chain rule of differentiation

8 The rules include:

( )d du dvu vdx dx dx

( )d du dvuv v udx dx dx

2( )

du dvv ud u dx dxdx v v

dy dy dudx du dx


p. 5


4.2 find the derivatives of algebraic functions, exponential functions and logarithmic functions

The formulae that students are required to use include:

( ) 0C

1( )n nx nx

( )x xe e

1ln )( xx

1(log )lna x

x a

( l) nx xa a a

Implicit differentiation and logarithmic differentiation are not required.

5. Second derivative

5.1 recognise the concept of the second derivative of a function

2 Students are required to recognise the

notations: y , ( )f x and 2

2d ydx

.

Third and higher order derivatives are not required.


p. 6


5.2 find the second derivative of an explicit function Students are required to recognise the second derivative tests and concavity.

6. Applications of differentiation

6.1 use differentiation to solve problems involving tangent, rate of change, maximum and minimum

10 Local and global extrema are required.

7. Indefinite integration and its applications

7.1 recognise the concept of indefinite integration

7.2 understand the basic properties of indefinite integrals and basic integration formulae

10 Indefinite integration as the reverse process of differentiation should be introduced.

Students are required to recognise the notation: ( )f x dx .

The properties include:

( ) ( )kf x dx k f x dx

( ) ( )f x g x dx

( ) ( )f x dx g x dx

The formulae include:

kdx kx C


p. 7


7.3 use basic integration formulae to find the indefinite integrals

of algebraic functions and exponential functions 7.4 use integration by substitution to find indefinite integrals 7.5 use indefinite integration to solve problems

1

1

nn xx dx C

n

1 ln xdxx

C

x xe dx e C

Students are required to understand the meaning of the constant of integration C. Integration by parts is not required.

8. Definite integration and its applications

8.1 recognise the concept of definite integration

12

The definition of the definite integral as the limit of a sum of the areas of rectangles under a curve should be introduced.


notation: ( )b

af x dx .


p. 8


8.2 recognise the Fundamental Theorem of Calculus and

understand the properties of definite integrals

The concept of dummy variables is required, for example,

( ) ( )b b

a af x dx f t dt .

The Fundamental Theorem of Calculus that students are required to recognise is:

( ) ( ) ( )b

af x dx F b F a , where

( ) ( )d F x f xdx

.


( ) ( )b a

a bf x dx f x dx

( ) 0a

af x dx

( ) ( ) ( )b c b

a a cf x dx f x dx f x dx

( ) ( )b b

a akf x dx k f x dx

( ) ( )b

af x g x dx


p. 9


8.3 find the definite integrals of algebraic functions and exponential functions

8.4 use integration by substitution to find definite integrals 8.5 use definite integration to find the areas of plane figures

8.6 use definite integration to solve problems

( ) ( )b b

a af x dx g x dx

Students are not required to use definite integration to find the area between a curve and the y-axis and the area between two curves.

9. Approximation of definite integrals using the trapezoidal rule

9.1 understand the trapezoidal rule and use it to estimate the values of definite integrals

4 Error estimation is not required.

Students are required to determine whether an estimate is an over-estimate or under-estimate by using the second derivative and concavity.


Statistics


p. 10


10. Conditional probability and Bayes’ theorem

10.1 understand the concept of conditional probability

6

10.2 use Bayes’ theorem to solve simple problems

11. Discrete random variables

11.1 recognise the concept of discrete random variables 1

12. Probability distribution, expectation and variance

12.1 recognise the concept of discrete probability distribution and represent the distribution in the form of tables, graphs and mathematical formulae

12.2 recognise the concepts of expectation E X and variance

Var( )X and use them to solve simple problems

7


( )E X xP X x

2Var( ) ( )X E X

( ) ( ) ( )E g X g x P X x

E aX b aE X b

2 2Var( ) ( )X E X E X

2Var( ) Var( )aX b a X


p. 11


13. The binomial distribution

13.1 recognise the concept and properties of the binomial distribution

5 The Bernoulli distribution should be introduced.

The mean and variance of the binomial distribution are required (the proofs are not required).

13.2 calculate probabilities involving the binomial distribution Use of the binomial distribution table is not required.

14. The Poisson distribution

14.1 recognise the concept and properties of the Poisson distribution

14.2 calculate probabilities involving the Poisson distribution

5 The mean and variance of Poisson distribution are required (the proofs are not required).

Use of the Poisson distribution table is not required.

15. Applications of the binomial and the Poisson distributions

15.1 use the binomial and the Poisson distributions to solve problems

5

16. Basic definition 16.1 recognise the concepts of continuous random variables and 3 Derivations of the mean and variance of


p. 12


and properties of the normal distribution

continuous probability distributions, with reference to the normal distribution

16.2 recognise the concept and properties of the normal distribution

the normal distribution are not required.

Students are required to recognise that the formulae in Learning Objective 12.3 are also applicable to continuous random variables.

The properties include: the curve is bell-shaped and

symmetrical about the mean the mean, mode and median are all

equal the flatness can be determined by the

value of the area under the curve is 1

17. Standardisation of a normal variable and use of the standard normal table

17.1 standardise a normal variable and use the standard normal table to find probabilities involving the normal distribution

2

18. Applications of 18.1 find the values of 1( )P X x , 2( )P X x , 1 2( )P x X x 7


p. 13


the normal distribution

and related probabilities, given the values of x1 , x2 , and , where 2~ ( , )X N

18.2 find the values of x, given the values of ( )P X x ,

( )P X x , ( )P a X x , ( )P x X b or a related probability, where 2~ ( , )X N

18.3 use the normal distribution to solve problems

19. Sampling distribution and point estimates

19.1 recognise the concepts of sample statistics and population parameters

19.2 recognise the sampling distribution of the sample mean X

from a random sample of size n

9 Students are required to recognise:

If the population mean is and the population size is N, then the population

variance is

2

2 1

( )N

ii

x

N

.

Students are required to recognise:

If the population mean is and the population variance is 2 , then

E X and 2

Var( )Xn

.


p. 14


19.3 use the Central Limit Theorem to treat X as being normally distributed when the random sample size n is sufficiently large

19.4 recognise the concept of point estimates including the sample

mean and sample variance

If 2~ ( , )X N , then 2

~ ( , )X Nn (the proof is not

required). Students are required to recognise:

If the sample mean is x and the sample size is n, then the sample

variance is

2

2 1

( )

1

n

ii

x xs

n

.

Students are required to recognise the concept of unbiased estimator.

20. Confidence interval for a population mean

20.1 recognise the concept of confidence interval

20.2 find the confidence interval for a population mean

6

Students are required to recognise:

A 100(1 )% confidence interval for the mean of a normal population with known variance 2 is


p. 15


given by2 2

( , )x z x zn n

.

When the sample size n is sufficiently large, a 100(1 )% confidence interval for the mean of a population with unknown variance

is given by 2 2

( , )s sx z x zn n

,

where s is the sample standard deviation.


Further Learning Unit

21. Inquiry and investigation

Through various learning activities, discover and construct knowledge, further improve the ability to inquire, communicate, reason and conceptualise mathematical concepts

7 This is not an independent and isolated learning unit. The time is allocated for students to engage in learning activities from different learning units.

Subtotal in hours 7

Grand total: 125 hours


p. 16

Learning Objectives of Module 2 (Algebra and Calculus) Notes:

1. Learning units are grouped under three areas (“Foundation Knowledge”, “Algebra” and “Calculus”) and a Further Learning Unit.

2. Related learning objectives are grouped under the same learning unit.

3. The notes in the “Remarks” column of the table may be considered as supplementary information about the learning objectives.

4. To aid teachers in judging how far to take a given topic, a suggested lesson time in hours is given against each learning unit. However, the lesson time assigned is for their reference only. Teachers may adjust the lesson time to meet their individual needs.


Foundation Knowledge

1. Odd and even functions

1.1 recognise odd and even functions and their graphs 2 Students are required to recognise that the absolute value function is an example of even functions.

2. Mathematical induction

2.1 understand the principle of mathematical induction 3 The First Principle of Mathematical Induction is required.

Students are required to prove propositions related to the summation of a finite sequence.

Proving propositions involving inequalities is not required.


p. 17


3. The binomial Theorem

3.1 expand binomials with positive integral indices using the binomial theorem

3 Proving the binomial theorem is required.

Students are required to recognise the summation notation ( ).

The following content are not required:

expansion of trinomials

the greatest coefficient, the greatest term and the properties of binomial coefficients

applications to numerical approximation

4. More about trigonometric functions

4.1 understand the concept of radian measure 4.2 understand the functions cosecant, secant and cotangent

13 The formulae that students are required to use include:

2 21 tan sec and 2 21 cot cosec

Simplifying trigonometric expressions by identities is required.


p. 18


4.3 understand compound angle formulae for the functions sine, cosine and tangent, and product-to-sum and sum-to-product formulae for the functions sine and cosine

The formulae include:

sin( ) sin cos cos sinA B A B A B

cos( ) cos cos sin sinA B A B A B

tan tantan( )1 tan tan

A BA BA B

2sin cos sin( ) sin( )A B A B A B

2 cos cos cos( ) cos( )A B A B A B

2sin sin cos( ) cos( )A B A B A B

sin sin 2sin cos2 2

A B A BA B

sin sin 2cos sin2 2

A B A BA B

cos cos 2cos cos2 2

A B A BA B

cos cos 2sin sin2 2

A B A BA B

5. Introduction to 5.1 recognise the definitions and notations of the number e and the 2 Two approaches for the introduction to


p. 19


the number e natural logarithm e can be considered:

1lim(1 )n

ne

n

(proving the existence of this limit is not required)

2 3

12! 3!

x x xe x

These definitions may be introduced in Learning Objective 6.1.


Calculus

6. Limits 6.1 understand the intuitive concept of the limit of a function

6.2 find the limit of a function

3 Student are required to recognise the theorems on the limits of sum, difference, product, quotient, scalar multiplication of functions and the limits of composite functions (the proofs are not required).



p. 20


0

sinlim 1

0

1lim 1x

x

ex

Finding the limit of a rational function at infinity is required.

7. Differentiation 7.1 understand the concept of the derivative of a function

7.2 understand the addition rule, product rule, quotient rule and chain rule of differentiation

14 Students are required to find the derivatives of elementary functions, including C , nx (n is a positive integer), x , sin x , cos x , xe , ln x from first principles.


notations: y , ( )f x and dydx

.

Testing differentiability of functions is not required.

The rules include:

( )d du dvu vdx dx dx


p. 21


7.3 find the derivatives of functions involving algebraic

functions, trigonometric functions, exponential functions and logarithmic functions

( )d du dvuv v udx dx dx

2( )

du dvv ud u dx dxdx v v

dy dy dudx du dx


( ) 0C

1( )n nx nx

(sin ) cosx x

(cos ) sinx x

2(tan ) secx x

( )x xe e

1ln )( xx

The following algebraic functions are required:


p. 22


7.4 find derivatives by implicit differentiation 7.5 find the second derivative of an explicit function

polynomial functions

rational functions

power functions x

functions formed from the above functions through addition, subtraction, multiplication, division and composition, such as 2 1x

Logarithmic differentiation is required. Students are required to recognise the

notations: y , ( )f x and 2

2d ydx

.

Students are required to recognise the second derivative tests and concavity.

Third and higher order derivatives are not required.

8. Applications of differentiation

8.1 find the equations of tangents to a curve 8.2 find the maximum and minimum value of a function

14

Local and global extrema are required.


p. 23


8.3 sketch curves of polynomial functions and rational functions

8.4 solve the problems relating to rate of change, maximum and minimum

The following points should be considered in curve sketching: symmetry of the curve limitations on the values of x and y intercepts with the axes maximum and minimum points points of inflexion vertical, horizontal and oblique

asymptotes to the curve

Students are required to deduce the equation of the oblique asymptote to the curve of a rational function by division.

9. Indefinite integration and its applications

9.1 recognise the concept of indefinite integration 9.2 understand the properties of indefinite integrals and use the

integration formulae of algebraic functions, trigonometric functions and exponential functions to find indefinite integrals

16 Indefinite integration as the reverse process of differentiation should be introduced. The formulae include:

k dx kx C


p. 24


9.3 understand the applications of indefinite integrals in mathematical contexts

9.4 use integration by substitution to find indefinite integrals

9.5 use trigonometric substitutions to find the indefinite integrals

involving 2 2a x , 2 2

1a x

or 2 21

x a

1

1

nn xx dx C

n

1 ln xdxx

C

x xe dx e C

sin cosx dx x C

cos sinx dx x C

2sec tanx dx x C

Applications of indefinite integrals in some fields such as geometry is required. Students are required to recognise the notations: sin1x , cos1x and tan1x , and their related principal values.


p. 25


9.6 use integration by parts to find indefinite integrals Teachers may use ln x dx as an

example to illustrate the method of integration by parts.

The use of integration by parts is limited to at most two times in finding an integral.

10. Definite integration

10.1 recognise the concept of definite integration

10.2 understand the properties of definite integrals

10

The definite integral as the limit of a sum and finding a definite integral from the definition should be introduced.

The concept of dummy variables is required, for example,

( ) ( )b b

a af x dx f t dt .

Using definite integration to find the sum to infinity of a sequence is not required. The properties include:

( ) ( )b a

a bf x dx f x dx

( ) 0a

af x dx


p. 26


10.3 find definite integrals of algebraic functions, trigonometric

functions and exponential functions

10.4 use integration by substitution to find definite integrals 10.5 use integration by parts to find definite integrals

( ) ( ) ( )b c b

a a cf x dx f x dx f x dx

( ) ( )b b

a akf x dx k f x dx

( ) ( )

( ) ( )

b

ab b

a a

f x g x dx

f x dx g x dx

( ) 0a

af x dx

if ( )f x is an odd

function

0

( ) 2 ( )a a

af x dx f x dx

if ( )f x is

an even function The Fundamental Theorem of Calculus that students are required to recognise is:

( ) ( ) ( )b

af x dx F b F a , where

( ) ( )d F x f xdx

.

The use of integration by parts is limited to at most two times in finding an integral.


p. 27


11. Applications of definite integration

11.1 understand the application of definite integrals in finding the area of a plane figure

11.2 understand the application of definite integrals in finding the volume of a solid of revolution about a coordinate axis or a line parallel to a coordinate axis

4

“Disc method” is required.


Algebra

12. Determinants 12.1 recognise the concept of determinants of order 2 and order 3 2 Students are required to recognise the notations: |A | and det A .

13. Matrices 13.1 understand the concept, operations and properties of matrices

10 The addition, scalar multiplication and multiplication of matrices are required.


A B B A

( ) ( )A B C A B C

( )A A A

( )A B A B


p. 28


13.2 understand the concept, operations and properties of inverses of square matrices of order 2 and order 3

( ) ( )A BC AB C

( )A B C AB AC

( )A B C AC BC

( )( ) ( )A B AB

AB A B


the inverse of A is unique

1 1( )A A

1 1 1( )A A

1 1( ) ( )n nA A

1 1( ) ( )T TA A

11A A

1 1 1( )AB B A

where A and B are invertible matrices and is a non-zero scalar.


p. 29


14. Systems of linear equations

14.1 solve the systems of linear equations in two and three variables by Cramer’s rule, inverse matrices and Gaussian elimination

6 The following theorem is required: A system of homogeneous linear equations has nontrivial solutions if and only if the coefficient matrix is singular

15. Introduction to vectors

15.1 understand the concepts of vectors and scalars

15.2 understand the operations and properties of vectors

5 The concepts of magnitudes of vectors, zero vector and unit vectors are required.

Students are required to recognise some common notations of vectors in printed form (including a and AB

) and in

written form (including a , AB

and a ) ; and some notations for magnitude (including a and a ). The addition, subtraction and scalar multiplication of vectors are required.


a b b a

( ) ( ) a b c a b c

a 0 a

0 a 0


p. 30


15.3 understand the representation of a vector in the rectangular

coordinate system

( ) ( ) a a

( ) a a a

( ) a b a b

If 1 1 a b a b (a and b are non-zero and are not parallel to each other), then 1 and 1 =


2 2 2OP x y z

in R3

2 2

sin yx y

and

2 2cos x

x y

in R2

The representation of vectors in the rectangular coordinate system can be used to discuss those properties listed in the Remarks against Learning Objective 15.2.

The concept of direction cosines is not


p. 31


required.

16. Scalar product and vector product

16.1 understand the definition and properties of the scalar product (dot product) of vectors

16.2 understand the definition and properties of the vector product

(cross product) of vectors in R3

5 The properties include:

a b b a

( ) ( ) a b a b

( ) a b c a b a c

2 0 a a a

0 a a if and only if a 0

a b a b

2 2 2 2( ) a b a b a b


a a 0

( ) b a a b

( ) a b c a c b c

( ) a b c a b a c

( ) ( ) ( ) a b a b a b


p. 32


2 2 2 2( ) a b a b a b

17. Applications of vectors

17.1 understand the applications of vectors 6 Division of a line segment, parallelism and orthogonality are required.

Finding angles between two vectors, the projection of a vector onto another vector and the area of a triangle are required.






Subtotal in hours 7



p. 1

The Learning Units for Key Stage 4 (S4 – S6)

Further Mathematics (Elective)

Foundation Knowledge Calculus Statistics Algebra

1. Odd and even functions

2. Mathematical induction

3. The binomial theorem

4. Exponential and

logarithmic functions

5. More about

trigonometric functions

6. Limits

7. Differentiation and its

applications

8. Indefinite integration

and its applications

9. Definite integration and

its applications

10. The trapezoidal rule

11. Conditional probability and Bayes’ theorem

12. Discrete random variables and probability

distributions

13. The binomial, the geometric and the

Poisson distributions, and their applications

14. The normal distribution and its applications

15. Sampling distribution and point estimates

16. Confidence interval for a population mean

17. Determinants

18. Matrices

19. Systems of linear

equations

20. Vectors and their

applications

21. Complex numbers




p. 2

Further Mathematics (Elective)


21. Complex numbers

21.1 understand the concepts and properties of the conjugate and modulus of a complex number

21.2 understand the polar form of a complex number

22 The properties include:

2zz z

z z

1 2 1 2z z z z

1 2 1 2z z z z

1 1

2 2

( )z zz z

1 2 1 2z z z z

11

2 2

zzz z

1 2 1 2z z z z

The real part (Re )z , imaginary part (Im )z , argument (arg )z and principal value of argument (Arg )z of a complex number z are required.

cisr as a short form of complex number


p. 3


21.3 perform multiplication and division of complex numbers in polar form

21.4 describe and sketch the locus of points satisfying

given conditions on an Argand diagram

(cos sin )r i should be introduced.

Students are required to represent complex numbers on an Argand diagram.

Students are required to convert any complex number z from standard form x yi to polar form (sin cos )r i and vice versa. Students are required to understand:

If 1 1 1 1(cos sin )z r and

2 2 2 2(cos sin )z r i , then

1 2 1 2 1 2 1 2cos( ) sin( )z z rr i and

1 11 2 1 2

2 2

cos( ) sin( )z r iz r

The conditions include:

1z z k

1 2z z z z

1arg( )z z


p. 4


21.5 understand de Moivre’s theorem and its applications

1

2

arg( )2

z zz z

or

Students are required to:

find nz , where n is an integer

find the nth roots of z

understand the cube roots of unity: 1 , , 2 and their properties 3 1 ,

21 0

solve problems related to trigonometric identities








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